Find the partial fraction decomposition for each rational expression. See Examples 1–4. (4x + 2)/((x + 2)(2x - 1))
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for performing operations like addition, subtraction, and decomposition. In this context, the expression (4x + 2)/((x + 2)(2x - 1)) is a rational expression that needs to be decomposed into simpler fractions.
Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions. This technique is particularly useful for integrating rational functions or simplifying complex expressions. The goal is to break down the given rational expression into components that are easier to work with, typically involving linear or irreducible quadratic factors.
Polynomial factorization involves breaking down a polynomial into its constituent factors, which can be linear or quadratic. This is essential for partial fraction decomposition, as the denominators of the simpler fractions must correspond to the factors of the original denominator. In the given expression, recognizing the factors (x + 2) and (2x - 1) is key to setting up the decomposition correctly.